American Institute of Mathematical Sciences

Advanced
May  2016, 15(3): 965-989. doi: 10.3934/cpaa.2016.15.965

Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials

Weiwei Ao 1, , Liping Wang 2, and  Wei Yao 3,

1. 

Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2
2. 

Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241
3. 

Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Chile

Received  October 2015 Revised  November 2015 Published  February 2016

Without any symmetric conditions on potentials, we proved the following nonlinear Schrödinger system \begin{eqnarray} \left\{\begin{array}{ll} \Delta u-P(x)u+\mu_1u^3+\beta uv^2=0, \quad &\mbox{in} \quad R^2\\ \Delta v-Q(x)v+\mu_2v^3+\beta vu^2=0, \quad &\mbox{in} \quad R^2 \end{array} \right. \end{eqnarray} has infinitely many non-radial solutions with suitable decaying rate at infinity of potentials $P(x)$ and $Q(x)$. This is the continued work of [8]. Especially when $P(x)$ and $Q(x)$ are symmetric, this result has been proved in [18].
Keywords: intermediate reduction method., non-symmetric potentials, infinitely many solutions, Schrödinger system.
Mathematics Subject Classification: Primary: 35B35, 35J25; Secondary: 35B40, 92D2.
Citation: Weiwei Ao, Liping Wang, Wei Yao. Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials. Communications on Pure & Applied Analysis, 2016, 15 (3) : 965-989. doi: 10.3934/cpaa.2016.15.965
References:
[1]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations,, \emph{C. R. Acad. Sci. Paris Ser.}, 1342 (2006), 453.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar

[2]

T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, \emph{Cal. Var. Partial Differential Equations.}, 37 (2010), 345.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[3]

T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, \emph{J. Fixed Point Theory Appl.}, 2 (2007), 353.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[4]

J. Y. Byeon and M. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations,, \emph{Memoirs of the American Mathematical Society, 229 (2014).   Google Scholar

[5]

K. Chow, Periodic solutions for a system of four coupled nonlinear Schrödinger equations,, \emph{Phys. Rev. Lett. A}, 285 (2001), 319.  doi: 10.1016/S0375-9601(01)00369-3.  Google Scholar

[6]

M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 19 (2002), 871.  doi: 10.1016/S0294-1449(02)00104-X.  Google Scholar

[7]

N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 27 (2010), 953.  doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[8]

M. del Pino, J. C. Wei and W. Yao, Intermediate reduction methods and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials,, \emph{Car. Var. PDE., 53 (2015), 473.  doi: 10.1007/s00526-014-0756-3.  Google Scholar

[9]

Y. Guo and J. Wei, Infinitely many positive solutions for nonlinear Schrödinger system with non-symmetric first order,, preprint., ().   Google Scholar

[10]

F. Hioe and T. Salter, Special set and solution of coupled nonlinear Schrödinger equations,, \emph{J. Phys. A: Math. Gen., 35 (2002), 8913.   Google Scholar

[11]

T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, \emph{Comm. Math Phys.}, 255 (2005), 629.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[12]

T. C. Lin and J. C. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations,, \emph{Phy. D}, 220 (2006), 99.  doi: 10.1016/j.physd.2006.07.009.  Google Scholar

[13]

Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger system,, \emph{Comm. math. Phys.}, 282 (2008), 721.  doi: 10.1007/s00220-008-0546-x.  Google Scholar

[14]

A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\R^N$,, \emph{Adv. Math.}, 221 (2009), 1843.   Google Scholar

[15]

M. Mitchell and M. Segev, Self-trapping of inconherent white light,, \emph{Nature}, 387 (1997), 880.  doi: 10.1038/43136.  Google Scholar

[16]

M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation,, \emph{J. Eur. Math. Soc.}, 14 (2012), 1923.  doi: 10.4171/JEMS/351.  Google Scholar

[17]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 267.   Google Scholar

[18]

S. J. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems,, \emph{Arch. Rational. Mech. Anal.}, 208 (2013), 305.  doi: 10.1007/s00205-012-0598-0.  Google Scholar

[19]

E. Timmermans, Phase seperation of Bose Einstein condensates,, \emph{Phys. Rev. Lett.}, 81 (1998), 5718.   Google Scholar

[20]

S. Terracini and G. Verzini, Multipulse phase in $k-$mixtures of Bose-Einstein condenstates,, \emph{Arch. Rat. Mech. Anal.}, 194 (2009), 717.  doi: 10.1007/s00205-008-0172-y.  Google Scholar

[21]

L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 4581.  doi: 10.1090/S0002-9947-10-04955-X.  Google Scholar

[22]

L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains,, \emph{Proc. Lond. Math. Soc.}, 102 (2011), 1099.  doi: 10.1112/plms/pdq051.  Google Scholar

[23]

L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2333.   Google Scholar

[24]

J. C. Wei and T. Weth, Nonradial symmetric bound states for system of two coupled Schrödinger equations,, \emph{Rend. Lincei Mat. Appl.}, 18 (2007), 279.   Google Scholar

[25]

J. C. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations,, \emph{Arch. Rat. Mech. Anal.}, 190 (2008), 83.  doi: 10.1007/s00205-008-0121-9.  Google Scholar

[26]

J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $R^n$,, \emph{Calc. Var. Partial Differential Equations}, 37 (2010), 423.  doi: 10.1007/s00526-009-0270-1.  Google Scholar

[27]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations,, \emph{CPAA}, 11 (2012), 1003.   Google Scholar

show all references

References:
[1]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations,, \emph{C. R. Acad. Sci. Paris Ser.}, 1342 (2006), 453.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar

[2]

T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, \emph{Cal. Var. Partial Differential Equations.}, 37 (2010), 345.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[3]

T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, \emph{J. Fixed Point Theory Appl.}, 2 (2007), 353.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[4]

J. Y. Byeon and M. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations,, \emph{Memoirs of the American Mathematical Society, 229 (2014).   Google Scholar

[5]

K. Chow, Periodic solutions for a system of four coupled nonlinear Schrödinger equations,, \emph{Phys. Rev. Lett. A}, 285 (2001), 319.  doi: 10.1016/S0375-9601(01)00369-3.  Google Scholar

[6]

M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 19 (2002), 871.  doi: 10.1016/S0294-1449(02)00104-X.  Google Scholar

[7]

N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 27 (2010), 953.  doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[8]

M. del Pino, J. C. Wei and W. Yao, Intermediate reduction methods and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials,, \emph{Car. Var. PDE., 53 (2015), 473.  doi: 10.1007/s00526-014-0756-3.  Google Scholar

[9]

Y. Guo and J. Wei, Infinitely many positive solutions for nonlinear Schrödinger system with non-symmetric first order,, preprint., ().   Google Scholar

[10]

F. Hioe and T. Salter, Special set and solution of coupled nonlinear Schrödinger equations,, \emph{J. Phys. A: Math. Gen., 35 (2002), 8913.   Google Scholar

[11]

T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, \emph{Comm. Math Phys.}, 255 (2005), 629.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[12]

T. C. Lin and J. C. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations,, \emph{Phy. D}, 220 (2006), 99.  doi: 10.1016/j.physd.2006.07.009.  Google Scholar

[13]

Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger system,, \emph{Comm. math. Phys.}, 282 (2008), 721.  doi: 10.1007/s00220-008-0546-x.  Google Scholar

[14]

A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\R^N$,, \emph{Adv. Math.}, 221 (2009), 1843.   Google Scholar

[15]

M. Mitchell and M. Segev, Self-trapping of inconherent white light,, \emph{Nature}, 387 (1997), 880.  doi: 10.1038/43136.  Google Scholar

[16]

M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation,, \emph{J. Eur. Math. Soc.}, 14 (2012), 1923.  doi: 10.4171/JEMS/351.  Google Scholar

[17]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 267.   Google Scholar

[18]

S. J. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems,, \emph{Arch. Rational. Mech. Anal.}, 208 (2013), 305.  doi: 10.1007/s00205-012-0598-0.  Google Scholar

[19]

E. Timmermans, Phase seperation of Bose Einstein condensates,, \emph{Phys. Rev. Lett.}, 81 (1998), 5718.   Google Scholar

[20]

S. Terracini and G. Verzini, Multipulse phase in $k-$mixtures of Bose-Einstein condenstates,, \emph{Arch. Rat. Mech. Anal.}, 194 (2009), 717.  doi: 10.1007/s00205-008-0172-y.  Google Scholar

[21]

L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 4581.  doi: 10.1090/S0002-9947-10-04955-X.  Google Scholar

[22]

L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains,, \emph{Proc. Lond. Math. Soc.}, 102 (2011), 1099.  doi: 10.1112/plms/pdq051.  Google Scholar

[23]

L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2333.   Google Scholar

[24]

J. C. Wei and T. Weth, Nonradial symmetric bound states for system of two coupled Schrödinger equations,, \emph{Rend. Lincei Mat. Appl.}, 18 (2007), 279.   Google Scholar

[25]

J. C. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations,, \emph{Arch. Rat. Mech. Anal.}, 190 (2008), 83.  doi: 10.1007/s00205-008-0121-9.  Google Scholar

[26]

J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $R^n$,, \emph{Calc. Var. Partial Differential Equations}, 37 (2010), 423.  doi: 10.1007/s00526-009-0270-1.  Google Scholar

[27]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations,, \emph{CPAA}, 11 (2012), 1003.   Google Scholar

[1]

Weiwei Ao, Juncheng Wei, Wen Yang. Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5561-5601. doi: 10.3934/dcds.2017242
[2]

Weiming Liu, Chunhua Wang. Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7081-7115. doi: 10.3934/dcds.2016109
[3]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094
[4]

Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427
[5]

Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071
[6]

Lushun Wang, Minbo Yang, Yu Zheng. Infinitely many segregated solutions for coupled nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6069-6102. doi: 10.3934/dcds.2019265
[7]

Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3407-3428. doi: 10.3934/dcdsb.2016104
[8]

Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure & Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004
[9]

Ziheng Zhang, Rong Yuan. Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials. Communications on Pure & Applied Analysis, 2014, 13 (2) : 623-634. doi: 10.3934/cpaa.2014.13.623
[10]

Dušan D. Repovš. Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 401-411. doi: 10.3934/dcdss.2019026
[11]

Veronica Felli, Alberto Ferrero, Susanna Terracini. On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3895-3956. doi: 10.3934/dcds.2012.32.3895
[12]

Haidong Liu, Zhaoli Liu. Positive solutions of a nonlinear Schrödinger system with nonconstant potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1431-1464. doi: 10.3934/dcds.2016.36.1431
[13]

Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure & Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003
[14]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51
[15]

Michael Röckner, Jiyong Shin, Gerald Trutnau. Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3219-3237. doi: 10.3934/dcdsb.2016095
[16]

Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329
[17]

Yuxia Guo, Jianjun Nie. Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6873-6898. doi: 10.3934/dcds.2016099
[18]

Andrzej Szulkin, Shoyeb Waliullah. Infinitely many solutions for some singular elliptic problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 321-333. doi: 10.3934/dcds.2013.33.321
[19]

Angela A. Albanese, Elisabetta M. Mangino. A class of non-symmetric forms on the canonical simplex of $\R^d$. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 639-654. doi: 10.3934/dcds.2009.23.639
[20]

Veronica Felli, Elsa M. Marchini, Susanna Terracini. On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 91-119. doi: 10.3934/dcds.2008.21.91

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences